The topological Cooper-pair pump and its applications to metrology

Friday 30 April 2010

Most physical systems depend on external parameters which, at special values, give additional symmetries to the system. At these points, quantum degeneracies may occur. When the parameters are easily tunable, one can explore quantum states not only at the degeneracy points but over the entire parameter space. Even when it is away from a symmetry point, the system is sensitive to the point’s presence. The reason is simple: at a degenerate point absolute quantum phases cannot be defined. From a topological point of view, a degeneracy is a defect in the wavefunction phase. This topology has important consequences such as the quantization of physical quantities (e.g. the quantum Hall effect).

For quantum circuits, the quantized quantity is the charge transferred when performing an adiabatic cycle by varying gate voltage and magnetic fl ux values. This quantization is geometric and does not depend on the precise shape of the cycles used. In this fashion, a quantized current can be generated. Such a current can be used in metrology as an electrical current standard, critical to the redefi nition of the International System of Units.

The circuit studied, a Cooper pair pump, is represented on Fig. 1. It has quantum degeneracies at isolated points in the three dimensional parameter space (φ=π[2π], ng1=2/3+p, ng2=2/3+q) and an integer topological charge can be associated to the phase defects at these points.

The most interesting consequence of the nontrivial topology is the quantization of the charge transferred through appropriately chosen cycles, which enclose completely the degeneracy points. The idea is to make periodic cycles covering densely a closed surface which encloses one or more degeneracies. The most convenient cycles are helices covering the surface of a cylinder of height 2π (the cylinder axis is the phase difference across the pump). Over one cycle, the charge transferred is an integer (the number of turns) multiple of 2e. When the cycles are covered at a frequency ν, the current Ip generated is exactly Ip=2epν. It can be used as a metrological current source and an absolute standard for the International Unit System. Figure 1: A Cooper pair pump has three Josephson junction in series defining two islands which electrostatic energies are tuned by two gate voltages. The phase difference across the pump is tuned by a magntic flux: the parameter space is here three-dimensional.

There are several challenges on the way. Since the current is generated through an adiabatic evolution in the circuit ground state, it is necessary to fulfill the appropriate adiabaticity criteria (freeze out Landau-Zener tunneling), which limit the pump frequency and hence the current to few tens op pico-ampere. The “topological” nature of the current generated protect the circuit against low frequency noise sources which do not degrade significantly the current accuracy. On the other hand, high frequency noise, can induce transition to other quantum states and may affect the pump accuracy. This research is part of a european program devoted to the adiabatic evolution of electronic circuits. Figure 2: A degenerate point in the three dimensional parameter space, is represented at the center (φ=π ) of the cylinder. The ellipses making p turns (p=2 on the figure) around the cylinder transfer p Cooper pairs through the pump.

Further reading : “The Cooper Pair Pump as a Quantized Current Source” , Raphael Leone, Laurent P. Lévy and Philippe Lafarge, Phys. Rev. Lett. 100, 117001 (2008) “Topological quantization by controlled paths: application to Cooper pairs pumps “, R. Leone, L. P. Lévy, Phys. Rev. B 77, 064524 (2008). Web site: Quantum coherence group

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